Crystalline materials have long been exploited in many optical and electronic applications for physical properties arising from their crystalline symmetry. Although such crystalline materials allow many technological applications to be fulfilled, there are limitations imposed by such crystalline symmetry. For example, ordered arrangements of dielectric materials with alternating domains of high and low index of refraction are known to exhibit a property for the transmission of light known as a photonic bandgap. The optical properties of a photonic bandgap material are characterized by a range of frequencies of light for which light cannot propagate, nor is it absorbed. This property is analogous to the electronic bandgaps that arise in semiconductors for the transport of electrons, and should result in a similarly broad spectrum of applications. The extent of a material's photonic bandgap depends both on the dielectric properties of the constituent dielectric materials and also on the symmetries of their three-dimensional arrangement. The limited set of distinct symmetries available for crystalline arrangements require a very large contrast in dielectric constant to achieve a full photonic bandgap, and these symmetries result in optical materials whose optical properties are very sensitive to structural and chemical defects. By contrast, quasicrystals are known that have far higher rotational symmetries than is possible for crystals. They consequently should exhibit larger and more uniform photonic bandgaps than any crystalline arrangement of the same materials, and should have optical properties that are more robust against defects and disorder. Two-dimensional and three-dimensional quasicrystalline arrangements of materials therefore should have a wide range of technological applications based on their optical and other physical properties.